The Geometry of Numbers (Anneli Lax New Mathematical Library, Series Number 41) 1st Edition by C. D. Olds (PDF)

Description

This is a self-contained introduction to the geometry of numbers, beginning with easily understood questions about lattice points on lines, circles and inside simple polygons in the plane. A minimum of mathematical expertise is required beyond an acquaintance with elementary geometry. The authors gradually lead up to the theorems of Minkowski and others who succeeded him. On the way the reader will see how this powerful approach gives improved approximations to irrational numbers by rationals, simplifies arguments on ways of representing integers as sums of squares, and provides a natural tool for attacking problems involving dense packings of spheres.

User’s Reviews

Editorial Reviews: Book Description A self-contained introduction to the geometry of numbers.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐item just as described; fast shipment, good packaging: A+

⭐Chapter 1-4 are dull. The authors try to introduce some basic concepts at a very elementary level, but instead of helping inexperienced readers this introduction will only bore them by being way to long and postponing all the really interesting material. In addition the exposition is just plain stupid at times. Let me illustrate by an example. On page 54 we learn of unique factorisation; for a proof we should “see Niven [7]”. This is bad enough as it is, but it is especially unacceptable since already on page 8 we proved the “fundamental theorem of arithmetic” that if p divides ab then it divides a or b, from where unique factorisation would follow trivially. Ironically, we are asked to make this very deduction in the case of Gaussian integers as an exercise in an appendix (p. 145). (Also, one wonders why we should “see Niven” in particular for this theorem that is proved in a million books, especially since for the Bolzano-Weierstass theorem we should “consult any advanced calculus textbook” (p. 16).)Chapters 5-10 are somewhat better. The authors finally get around to presenting the basics of Minkowski’s theory, especially his “fundamental theorem” that any centrally symmetric, convex figure with area > 4 contains several lattice points. There are three main applications:1. Diophantine approximation, i.e. the problem of finding rational approximations to irrational numbers that are as good as possible relative to the size of the denominator.2. Quadratic forms, in particular finding bounds for the minimum value attained by a quadratic form.3. The theorem that any integer is the sum of four squares.After their discussion of quadratic forms, our authors insert a section 8.5 where they note that these ideas “can be applied to the problem of approximating irrational numbers by rational numbers”, and then they lunge into a discussion, apparently unaware that this is just what they did 30 pages ago, in section 6.4. And in a final display of poor craftsmanship, most page references in the index are off by several pages.

⭐This choppy patchwork presentation suffered throughlong gestation and multiple authorship resulting unfortunately inan untidy mess. It would have been better simply to abort the wholeproject or else bring in someone who is actually an expert in this area.However, in that case it would have been necessary to start all overagain anyway, so there was simply no hope for it but to bail out.Too bad for all of us since the topic is one of genuine interest and, incapable hands, could really appeal to motivated undergraduates.

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